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Solution - Absolute value equations

Exact form:

z = 11, - \frac{1}{3}

z=11 , -\frac{1}{3}

Decimal form:

z = 11, - 0.333

z=11 , -0.333

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| 5 z - 4 | = | 4 z + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 z - 4 \| = \| 4 z + 7 \|$
$x = + y$	$(5 z - 4) = (4 z + 7)$
$x = - y$	$(5 z - 4) = - (4 z + 7)$
$+ x = y$	$(5 z - 4) = (4 z + 7)$
$- x = y$	$- (5 z - 4) = (4 z + 7)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| 5 z - 4 \| = \| 4 z + 7 \|$
$x = + y$ , $+ x = y$	$(5 z - 4) = (4 z + 7)$
$x = - y$ , $- x = y$	$(5 z - 4) = - (4 z + 7)$

2. Solve the two equations for z

7 additional steps

$(5 z - 4) = (4 z + 7)$

Subtract from both sides:

$(5 z - 4) - 4 z = (4 z + 7) - 4 z$

Group like terms:

$(5 z - 4 z) - 4 = (4 z + 7) - 4 z$

Simplify the arithmetic:

$z - 4 = (4 z + 7) - 4 z$

Group like terms:

$z - 4 = (4 z - 4 z) + 7$

Simplify the arithmetic:

$z - 4 = 7$

Add to both sides:

$(z - 4) + 4 = 7 + 4$

Simplify the arithmetic:

$z = 7 + 4$

Simplify the arithmetic:

$z = 11$

12 additional steps

$(5 z - 4) = - (4 z + 7)$

Expand the parentheses:

$(5 z - 4) = - 4 z - 7$

Add to both sides:

$(5 z - 4) + 4 z = (- 4 z - 7) + 4 z$

Group like terms:

$(5 z + 4 z) - 4 = (- 4 z - 7) + 4 z$

Simplify the arithmetic:

$9 z - 4 = (- 4 z - 7) + 4 z$

Group like terms:

$9 z - 4 = (- 4 z + 4 z) - 7$

Simplify the arithmetic:

$9 z - 4 = - 7$

Add to both sides:

$(9 z - 4) + 4 = - 7 + 4$

Simplify the arithmetic:

$9 z = - 7 + 4$

Simplify the arithmetic:

$9 z = - 3$

Divide both sides by :

$\frac{(9 z)}{9} = \frac{- 3}{9}$

Simplify the fraction:

$z = \frac{- 3}{9}$

Find the greatest common factor of the numerator and denominator:

$z = \frac{(- 1 \cdot 3)}{(3 \cdot 3)}$

Factor out and cancel the greatest common factor:

$z = \frac{- 1}{3}$

3. List the solutions

$z = 11, - \frac{1}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 z - 4 |$
$y = | 4 z + 7 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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